Scientia Iranica E (2015) 22(6), 2716{2721

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringwww.scientiairanica.com

Research Note

A mathematical model to evaluate knowledge in theknowledge-based organizations

R. Bagheria;�, A. Rezaeiana and A. Fazlalyb

a. Faculty of Management and Accounting, Shahid Beheshti University, Daneshju Blvd, Evin Square, Tehran, P.O. Box1983963113, Iran.

b. Faculty of Industrial Engineering, Khajehnasir Toosi University, Kaviyan, Tehran, P.O. Box 1541849611, Iran.

Received 24 September 2012; accepted 11 October 2014

KEYWORDSKnowledgemanagement;Data envelopmentanalysis;Mathematical model;BCC model;Fuzzy set.

Abstract. Knowledge and its intangible appurtenances have not only resulted inmovement in various businesses, but also, nowadays, have been viewed as the whole ora part of products of distributors' companies as well as service and military organizations.In recent years, estimation of knowledge level in organizations and industry companies hasattracted considerable attentions. Contrary to a lot of prevalent models used for measuringe�ciency, Data Envelopment Analysis (DEA) can take into account multiple inputs andoutputs. In this regard, DEA was traditionally applied with crisp inputs and outputs,while in practical cases, we need to estimate organization e�ciency in a di�erent situationin which we have to deal with fuzzy or imprecise data. The aim of this paper is to presenta DEA, employing fuzzy input and output data, toward assessing the knowledge levelsestablished in a knowledge based-organization in various time intervals. In this case, theorganization is able to de�ne some areas in which it can improve its established knowledgelevel.

© 2015 Sharif University of Technology. All rights reserved.

1. Introduction

Nowadays, companies feel more pressure resulted fromcompetitive market than before. In the era in whicheconomy is going to be smaller, they should employfewer resources to do more work. In order to establishnew knowledge, many organizations have recently beeninclined to hire Knowledge Management (KM) sys-tems. Some specialists believe that the recent obtainedknowledge may only be the result of this competition.By using the obtained knowledge, a better knowledgemanagement system is made, a more e�cient team

*. Corresponding author. Tel.: +98 21 22431842E-mail addresses: R Bagheri@Sbu.ac.ir (R. Bagheri);A Rezaeian@Sbu.ac.ir (A. Rezaeian); fazlaly@gmail.com(A. Fazlaly)

working is constituted, and �nally learning is necessary.The concept of knowledge does not necessarily meaninformation transferring, but creativity and dynamism.In this case, a knowledge-based organization is ideamaker, applies new ideas, and hence achieves competi-tive merit.

Shang & Sueyoshi [1] introduced DEA as a linearprogramming based technique to measure relative ef-�ciency of an organization or di�erent organizations,such as banking, post, hospitals, education centers,powerhouses, re�neries, and so on. In addition toe�ciency score, DEA can give very useful results tomanagers and determine pioneers for ine�cient DMUs.This technique also provides strategic solution ande�ciency improvement for developing the organization,and �nally assists organizations to assign resourcesproperly [2].

R. Bagheri et al./Scientia Iranica, Transactions E: Industrial Engineering 22 (2015) 2716{2721 2717

This paper attempts to propose DEA-based ap-proach with fuzzy data to estimate relative e�ciency ofknowledge-based organizations using knowledge man-agement criteria. In this case, the organizations areable to detect their improvement areas.

The rest of this paper is organized as follows:Section 2 gives a brief literature review. The proposedmathematical model is introduced in Section 3. Inputsand outputs of the proposed DEA model are de�ned inSection 4. The solution procedure and computationalresults are given in Section 5, and �nally Section 6 isdevoted to some conclusion remarks.

2. Literature review

American Productivity and Quality Center (APQC)de�nes knowledge as strategies and processes fordetecting, acquiring, and distributing knowledge tostrengthen the competition power of the organization.Based on the idea of Gartner Group, many organiza-tions are going to implement knowledge managementsystems and one third of the fortune 1000 companiesapply knowledge management plans and strategies.Davenport & Prusak [3] introduces knowledge man-agement as a structured approach containing proce-dures for detecting, evaluating, organizing, saving, andimplementing knowledge toward covering requirementsand objectives of organizations. Jakov Crnkovic etal. [4] proposed a comprehensive approach to KMwhich relates key processes of KM to critical factorsfor implementing it successfully. Using a matrixcomposing of the process and key success factors,they described an organization and estimated the KMcriteria.

E�ciency evaluation has been comprehensivelyaddressed by scholars because of its importance forperformance evaluation of an organization.

Farrell [5] exploited a method which had beenemployed for e�ciency evaluation, in engineering is-sues, to measure e�ciency of a production trim. Thestudy by Farrell involved just one input and oneoutput. Farrell's study included measuring technicale�ciency and derivative of e�cient production func-tion. Data Envelopment Analysis (DEA) is a mathe-matical programming based approach to estimate tech-nical e�ciency and ine�ciencies in production. Thismethod does not suppose any pre-determined shapefor production function curve and estimates productionor cost function applying mathematical models foractual input and output data of Decision Making Units(DMUs).

Charnes et al. [6] proposed the �rst mathematicalDEA based on Farrell's de�nition of relative e�ciency.The aim of this model was to measure and comparerelative e�ciency of Decision Making Units (DMUs)using multiple homogeneous inputs and outputs.

Banker et al. [7] proposed a new DEA modelcalled BCC, which considered e�ciency of DMUs withvariable return to scale.

In fuzzy environment, Sengupta [8] proposed afuzzy programming approach in which the objectivefunction was crisply, not satis�ed. He examined DEAmodel with multiple inputs and one output.

Kao and Liu [9] developed a method to �nd themembership functions of the fuzzy e�ciency scoreswhen some observations are in fuzzy numbers. En-tani et al. [10] proposed interval e�ciency obtainedfrom the pessimistic and the optimistic viewpoints.Since Zadeh [11,12] initiated the possibility measure,many researchers such as Guo & Tanaka [13] andS. Lertworasirikul et al. [14] have introduced it intoDEA.

Wen & Li [15] proposed a fuzzy DEA model forranking and evaluating. In this regard, they introduceda hybrid model with fuzzy simulation and geneticalgorithm, in which fuzzy triangular and trapezoidalfuzzy number were exploited. They, then, transformedthe hybrid model to a linear mathematical model andapplied it to solve a simple numerical example.

Yuan Feng Wen [16] introduced criteria to mea-sure knowledge management by reviewing literature,using scholars and practitioners in the areas of KM andAHP and a questionnaire. Leibowitz & Suen [17] andBose [18] proposed more criteria to measure knowledgemanagement. There exist many criteria to measureKM; the interesting point is that the majority ofthe proposed criteria are human based and can beutilized to quantify knowledge management plans andimplementation. The criteria proposed by ICM groupinclude value adding customer resource, infrastructureresource and human resource the proposed mathemat-ical model.

In this section, a fuzzy DEA mathematical modelis formulized to assess knowledge level of a knowledge-based organization in various time intervals, whichcan monitor the organization progress in terms ofknowledge management. In this regard, the BCCmodel is developed by fuzzy input and output datausing Zadeh's extensions principle. The followingprogramming problem gives the upper band of �-levelcut of fuzzy e�ciency measure of a DMU understudy,in which the primal BCC model is applied:

MaxZ0 =sXr=1

�ryr0 + w0;

sXr=1

�ryrj + w0 �mXi=1

wixij � 0;mXi=1

wixi0 = 1;

2718 R. Bagheri et al./Scientia Iranica, Transactions E: Industrial Engineering 22 (2015) 2716{2721

(~yrj)L� � yrj � (~yrj)U� ;

(~xij)L� � xij � (~xij)U� ;

�r; wi � 0; w0 free:In the above-mentioned programming problem, xijdenotes the ith input of the jth DMU. The weightassigned to the ith is represented by wi. On the otherhand, yrj is the rth output from the jth DMU. Theweight of rth output is described by �r. The variablereturn to scale in the aforementioned programmingproblem is represented by w0, where if w0 < 0, then itis said that return to scale is decreasing, and if w0 > 0,then it is increasing, otherwise, it is constant. Since,in the present study, it is supposed that the inputand output data are in the form of fuzzy numbers, �-level cut of ~xij and ~yrj are represented as (~xij)� =h(~xij)

L� ; (~xij)

U�

iand (~yrj)� =

h(~yrj)

L� ; (~yrj)

U�

i, respec-

tively.In order to obtain the lower bound of �-level

cut of fuzzy e�ciency measure, the dual form of BSSmodel is applied. In this case, using Zadeh's extensionprinciple, we have:

Miny0 = �;

nXj=1

�jyrj � yr0;

�xi0 �nXj=1

�jxij � 0;

nXj=1

�j = 1;

(~yrj)L� � yrj � (~yrj)U� ;

(~xij)L� � xij � (~xij)U� ;

�j � 0; � free:In the aforementioned programming problem, �and �j , j = 1; � � � ; n, are the decision vari-ables corresponding = and � constraints, respec-tively, in primal BCC model. Other parametersand variables are the same as those in primal BCCmodel.

3. Introducing input and output in theproposal model

It should be pointed out that DEA models are su�-ciently valid if their inputs and outputs are identi�edproperly. This has a signi�cant role for model's valida-tion. In the present study, to assess knowledge level bythe mathematical models, several criteria are de�nedusing experts' opinion and reviewing the literature,which directly relate to knowledge management andinuence knowledge level of the organization. Inthis regard, inputs and outputs to be used in theproposed fuzzy DEA model are de�ned as representedin Table 1.

4. Solution procedure

In this section, input and output data collected fromreviewing literature are introduced to be used in theproposed fuzzy BCC model. Table 2 represents thesedata for four various time intervals that includes theinput data of the processes (min) and the amountof investment (million), respectively. The fourth,�fth, and sixth columns are the output data which

Table 1. Inputs and outputs of the proposed model.

Inputs Outputs

Amount of investment Knowledge sharing among personnel

Average time of each process The number of software and new plans

Table 2. The inputs and outputs data of the problem.

Timeinterval

The averagetime of

processes

The amountof investment

The numberof processes

without error

Knowledgesharing

The numberof software

1 (20,24,27) (120,148,155) (14,17,22) (46,53,55) (7,9,11)

2 (18,21,24) (132,140,146) (14,18,20) (44,48,51) (5,7,11)

3 (20,26,28) (134,144,150) (10,14,17) (47,56,59) (6,10,13)

4 (15,20,26) (145,151,153) (16,20,23) (47,50,56) (6,13,19)

R. Bagheri et al./Scientia Iranica, Transactions E: Industrial Engineering 22 (2015) 2716{2721 2719

indicate the number of processes without error, knowl-edge sharing, and the number of software, respec-tively.

5. Computational result

In this section, the proposed fuzzy BCC model is solvedfor a knowledge based organization with the data givenin Table 2 and Lingo 12 solver. The outcome of theproposed model, in fact �-level cut of fuzzy e�ciencymeasure of the organization under study, is applied toconstruct membership functions, depicted in Figures 1to 4.

After deriving the membership functions fuzzye�ciency measures as described before, it is the turnto describe a procedure to be used for making adecision about whether or not the capability of thevariable measurement system under study with fuzzydata is acceptable. To do this, the abovementionedcriteria should be compared with their correspondingcritical values in fuzzy environment. In this regard,it necessitates a method to be applied for rankingfuzzy numbers. There are a lot of methods in theliterature in this area [19-23]. However, most ofthem need membership functions of the fuzzy numbersto be ranked. The method proposed by Chen andKlein [20], on the other hand, is very appropriatefor the present study, because it is based on �-levelcuts.

To describe the method proposed by Chen andKlein [20], suppose the fuzzy numbers ~Aj ; j =1; � � � ;m, are going to be ranked. Let h be themaximum height of � ~Aj ; j = 1; � � � ;m. Assume his split into n equal intervals such that �i = ih=n;i = 0; � � � ; n. Chen and Klein [20] proposed thefollowing index to be used for ranking fuzzy numbers:

Ij =nXi=1

��~Aj�U�i� c�

,"nXi=1

��~Aj�U�i� c��

nXi=1

��~Aj�L�i� d�#

n!1;where:

c = mini;j

��~Aj�L�i

�;

d = maxi;j

��~Aj�U�i

�:

The larger (smaller) value of index Ij , the larger(smaller) the fuzzy number ~Aj is. While this method is

Figure 1. Membership function of fuzzy e�ciency scorein the �rst time interval.

Figure 2. Membership function of fuzzy e�ciency scorein the second time interval.

Figure 3. Membership function of fuzzy e�ciency scorein the third time interval.

2720 R. Bagheri et al./Scientia Iranica, Transactions E: Industrial Engineering 22 (2015) 2716{2721

Figure 4. Membership function of fuzzy e�ciency scorein the fourth time interval.

authentic when n advances in�nity, Chen and Klein [20]proposed that n = 3 or 4 su�ces to discriminate thedi�erences.

Using the abovementioned procedure for rankingfuzzy e�ciency measure of the organization in four suc-cessive time intervals with the membership functionsdelineated in Figures 1-4, we have c = 0:0189 andd = 1. In this case, Chen and Klein's index for obtainedfuzzy e�ciency measures is corrupted as follow:

IDMU1 = 0:6976; IDMU2 = 0:7374;

IDMU3 = 0:5579; IDMU4 = 0:8535:

As it is observed, the organization is the most andleast e�cient in the fourth and third time intervals,respectively.

6. Conclusion

This paper was an e�ort to propose a fuzzy BCC modelto assess knowledge level of a knowledge-based organi-zation, wherein it was supposed that input and outputdata are fuzzy numbers. The proposed approach canalso be used to rank similar organizations in termof knowledge management criteria. The approach isspeci�cally applicable for organizations which want tocompare themselves with similar organizations in termsof knowledge management and select the most e�cientbenchmark.

In this case, the ine�cient organizations try toimprove their performance in the area of knowledgemanagement toward being assessed as e�cient.

References

1. Shang, J. and Sueyoshi, T. \A uni�ed frameworkfor the selection of a exible manufacturing system",European Journal of Operational Research, 85(2), pp.297-315 (1995).

2. Brown, M.G. and Svenson, R.A. \Measuring R &D productivity", Research Technology Management,41(6), pp. 30-35 (1998).

3. Davenport, T.H. and Prusak, L., Working Knowledge:Managing What Your Organization Knows, HarvardBusiness School Press (1998).

4. Crnkovic, J., Belardo, S. and Asoh, D.A. \Exploringthe knowledge management index as a performancediagnostic tool", Systemics, Cybernetics and Informat-ics, 3(2), pp. 27-33 (2004).

5. Farrel, M.J. \The measurement of productive e�-ciency", Journal of the Royal Statistical Society, 120,pp. 253-281 (1957).

6. Charnes, A., Cooper, W.W. and Rhodes, E. \Measur-ing the e�ciency of decision making units", EuropeanJournal of Operational Research, 2, pp. 429-444 (1978).

7. Banker, R.D., Charnes, A. and Cooper, W.W. \Somemodels for estimating technical and scale e�cienciesin data envelopment analysis", Management Science,30(9), pp. 1078-1092 (1984).

8. Sengupta, J.K. \A fuzzy system approach in data en-velopment analysis", Computers & Mathematics withApplications, 24, pp. 259-266 (1992).

9. Kao, C. and Liu, S.T. \Fuzzy e�ciency measures indata envelopment analysis", Fuzzy Sets and Systems,119, pp. 149-160 (2000).

10. Entani, T., Maeda, Y. and Tanaka, H. \Dual modelsof interval DEA and its extension to interval data",European Journal of Operational Research, 136(1), pp.32-45 (2002).

11. Zadeh, L.A. \Fuzzy sets", Information and Control, 8,pp. 338-353 (1965).

12. Zadeh, L.A. \Fuzzy sets as a basis for a theory ofpossibility", Fuzzy Sets and Systems, 1, pp. 3-28(1978).

13. Guo, P. and Tanaka, H. \Fuzzy DEA: A perceptualevaluation method", Fuzzy Sets and Systems, 119, pp.149-160 (2001).

14. Lertworasirikul, S., Fang, S.C., Joines, J.A. and Nut-tle, H.W. \Fuzzy data envelopment analysis (DEA):A possibility approach", Fuzzy Sets and Systems, 139,pp. 379-394 (2003).

15. Wen, M. and Li, H. \Fuzzy data envelopment analysis(DEA): Model and ranking method", Journal of Com-putational and Applied Mathematics, 223, pp. 872-878(2009).

16. Wen, Y.F. \An e�ectiveness measurement model forknowledge management", Knowledge-Based Systems,22, pp. 363-367 (2009).

17. Leibowitz, J. and Suen, C.Y. \Developing knowledgemanagement metrics for measuring intellectual capi-tal", Journal of Intellectual Capital, 1(1), pp. 54-67(2000).

18. Bose, R. \Knowledge management metrics", IndustrialManagement & Data Systems, 104(6), pp. 457-468(2004).

19. Asady, B. and Zendehnam, A. \Ranking fuzzy num-

R. Bagheri et al./Scientia Iranica, Transactions E: Industrial Engineering 22 (2015) 2716{2721 2721

bers by distance minimization", Applied MathematicalModelling, 31(11), pp. 2589-2598 (2007).

20. Chen, C.B. and Klein, C.M. \A simple approach toranking a group of aggregated fuzzy utilities", IEEETransaction on Systems Man and Cybernetics, 27(1),pp. 26-35 (1997).

21. Chou, S.Y., Dat, L.Q. and Yu, V.F. \A revised methodfor ranking fuzzy numbers using maximizing set andminimizing set", Computers & Industrial Engineering,61(4), pp. 1342-1348 (2011).

22. Chu, T.C. and Tsao, C.T. \Ranking fuzzy numberswith an area between the centroid point and originalpoint", Computers and Mathematics with Applica-tions, 43, pp. 111-117 (2002).

23. Wang, X. and Kerre, E.E. \Reasonable properties forthe ordering of fuzzy quantities (I) and (II)", FuzzySets and Systems, 118(3), pp. 387-405 (2001).

Biographies

Rouhollah Bagheri was born on March 1981, in Iran.He received his Master's degree in MBA from Amirk-abir University of Technology, Tehran, Iran, in 2012,and he is currently a PhD student in Systems Manage-ment at the Faculty of Management and Accounting inShahid Beheshti University. He has published 9 booksand 12 international papers in di�erent conferences

and journals; also, he has won 5 international andnational awards. His major research interests includeinformation technology management, business processreengineering, knowledge engineering, arti�cial intelli-gent methods, and knowledge management, especiallyknowledge strategies.

Ali Rezaeian received his PhD in Organizational Be-havior from United States International University, SanDiego, California. He is now president at the Facultyof Management and Accounting in Shahid BeheshtiUniversity. He became a member of Iranian Scienceand Culture Hall of Fame in the �eld of Management in2006. He has published a lot of books and internationalpapers in di�erent conferences and journals. His ma-jor research interests include organizational behavior,information technology management, and knowledgemanagement, especially knowledge strategies.

Amir Fazlaly was born on March 1983, in Iran. Hereceived his Master's degree in Industrial Engineer-ing from Khajehnasir Toosi University of Technology,Tehran, Iran, in 2012. He is currently a researcher ata research �rm. His major research interests include,business process reengineering, knowledge engineer-ing, and knowledge management, especially knowledgestrategies.